Hamilton-Poisson formulation for the rotational motion of a rigid body in the presence of an axisymmetric force field and a gyroscopic torque

TL;DR

This paper establishes conditions under which the rotational motion of a rigid body with an axisymmetric force field and gyroscopic torque can be described using Hamilton-Poisson structures, recovering conserved quantities through modifications.

Contribution

It introduces a framework for Hamilton-Poisson formulation of rigid body dynamics with gyroscopic effects, including conditions for conservation laws and Casimir functions.

Findings

Recovered Casimir functions for modified Poisson structures.

Applied framework to classical rigid body problems.

Provided sufficient conditions for Hamilton-Poisson formulation.

Abstract

We give sufficient conditions for the rigid body in the presence of an axisymmetric force field and a gyroscopic torque to admit a Hamilton-Poisson formulation. Even if by adding a gyroscopic torque we initially lose one of the conserved Casimirs, we recover another conservation law as a Casimir function for a modified Poisson structure. We apply this frame to several well known results in the literature.